02/18/2020

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- The following topics will be covered in this lecture:
- Events
- Sample spaces
- Classical version of probability
- Relative frequency approximation of probability
- Probabilistic reasoning
- Complements of events
- Odds

In order to interpret a

**population**from a**sample**, we required that the sample was**representative of the full population**.- Using random chance to mix the participants in the samples was one way of ensuring that the smaller
**sample**would be a**good approximation**of the**population**.

- Using random chance to mix the participants in the samples was one way of ensuring that the smaller
However, in any

**sample**there is natural variation amongst those sampled that leads to**two repeated samples not to look like eachother**.- One of the important considerations is thus, how likely would it be to compute a
**sample statistic**just by chance, due to the natural variation of resampling?

- One of the important considerations is thus, how likely would it be to compute a
**Inferential statistics**are differentiated from the**descriptive statistics**we have seen so far in how they address this above question.- Descriptive statistics help us learn about the sample we have in hand, but we must use tools from
**probability to address how these results might generalize to a wider population**.

- Descriptive statistics help us learn about the sample we have in hand, but we must use tools from
One of the most important tools we will learn in this class is

**hypothesis testing**, as one way to test if a claim might be infered about the wider population.

Courtesy of Mario Triola, *Essentials of Statistics*, 6th edition

- We can take a basic example to illustrate this point.
- Suppose a clinical trial will be used to determine if a certain fertility treatment will increase the chance that a pregnancy will result in a female birth.
- There is some random chance involved in an un-assisted pregnancy whether the baby will be a girl or boy, and either result is about as likely as the other.
- Therefore, every group of \( 100 \) births in a control group might look quite different.
- We want to find a way to be more confident that if there is an effect of the treatment, it can be distinguished from this random variation.
- Formally we will make a
**claim**and a**hypothesis**: **Claim**: the fertility treatment will greatly increase the chance of a baby being born a girl over the control group.**Hypothesis**: the treatment has no effect and it is equally likely that a pregnancy will result in a female or a male birth under the treatment.

**Note**: in the above, we take the**null hypothesis**, i.e.,**to test the claim we assume that it is not true and then evaluate the results**.